Quantum Entanglement, Locality, Determinism

Introduction

The journey into the world of quantum mechanics often begins with one of the most famous experiments in the history of physics: the Double-slit experiment. This experiment, first conducted by Thomas Young in 1801, was originally designed to demonstrate the wave nature of light. However, when extended to quantum objects like electrons, it revealed one of the most profound mysteries of the quantum world.

The Double-slit Experiment: A Historical Insight

In Young’s original experiment, light was shone through two closely spaced slits, and the resulting pattern was observed on a screen behind the slits. Rather than producing two distinct bright spots corresponding to the slits, the light formed an interference pattern—a series of bright and dark fringes—indicating that light behaves like a wave, with the waves from the two slits interfering with each other.

Fast forward to the early 20th century, and physicists began to wonder what would happen if they performed the same experiment with particles like electrons instead of light. Given that electrons were thought to be particles, it was expected that they would behave like tiny bullets, passing through one slit or the other and creating two distinct impact points on the screen. But what they observed was startling: when electrons were fired one at a time, they still formed an interference pattern on the screen, just as if they were waves.

The Mystery of Superposition

This result led to the realization that quantum objects, such as electrons, do not behave like classical particles. Instead, each electron seemed to pass through both slits simultaneously, existing in a superposition of going through the left slit and the right slit. The superposition was only resolved when the electron was measured—at which point it appeared to have gone through just one slit, and the interference pattern vanished.

Mathematically, this idea of superposition is described as:

$ |\psi\rangle = \alpha|L\rangle + \beta|R\rangle $

Here, $|L\rangle$ represents the state where the electron goes through the left slit, and $|R\rangle$ represents the state where it goes through the right slit. The coefficients $\alpha$ and $\beta$ are complex numbers, and the squares of their magnitudes give the probabilities of the electron being found in one state or the other when measured.

The Double-slit experiment not only demonstrated the wave-particle duality of quantum objects but also introduced the concept of quantum superposition. Before measurement, quantum objects like electrons exist in a superposition of all possible states. It is only when we observe or measure the system that it "chooses" a definite outcome.

From Superposition to Entanglement

In the classical world, the simplest unit of information is a bit, which can take on one of two possible values: 0 or 1. These bits are the foundation of all classical computing, with physical realizations ranging from early electromechanical devices and vacuum tubes to the transistors in modern computers. Essentially, different voltage levels in a circuit are used to represent the binary states of 0 and 1.

However, in the quantum world, the analogue of the bit is the qubit. A qubit can also take on values 0 or 1 when measured, but before measurement, quantum mechanics allows it to exist in a superposition of both states simultaneously. Mathematically, this superposition is written as:

$ |\psi\rangle = \alpha|0\rangle + \beta|1\rangle $

Here, $\alpha$ and $\beta$ are complex numbers (which have both a magnitude and a phase), and their squares $|\alpha|^2$ and $|\beta|^2$ represent the probabilities of measuring the qubit as 0 or 1, respectively. Physical realizations of qubits vary, but common examples include the spin of electrons or the polarization of photons. These quantum systems can be manipulated to exhibit the full range of quantum phenomena, including superposition and, more importantly, entanglement.

Scaling Up: From One Qubit to Two

Let’s imagine a quantum system that can be in one of four possible states, which we’ll label 0, 1, 2, and 3. If this system is composed of two qubits, we can describe the system's state as a superposition of four possibilities:

$ |\psi\rangle = \alpha|00\rangle + \beta|01\rangle + \delta|10\rangle + \gamma|11\rangle $

In this notation, $|00\rangle$ represents both qubits being in the state 0, $|01\rangle$ represents the first qubit in state 0 and the second in state 1, and so on. The coefficients $\alpha$, $\beta$, $\delta$, and $\gamma$ are complex numbers whose squares give the probabilities of each possible outcome when the system is measured.

Now, suppose we prepare the two-qubit system in a specific quantum state:

$ |\psi\rangle = \frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle $

This state is a special kind of quantum superposition known as an entangled state. Here, the two qubits are not independent; their outcomes are correlated, meaning that the measurement result of one qubit is directly related to the measurement result of the other qubit.

The EPR Experiment and Spacelike Separation

To illustrate the peculiarities of quantum mechanics, let’s consider a thought experiment proposed by Einstein, Podolsky, and Rosen, commonly known as the EPR experiment:

1. Preparation: Prepare a two-qubit system in the entangled state $\frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle$.
2. Separation: Physically separate the two qubits and send them to two distant locations, where Alice and Bob will measure them.
3. Measurement: Alice and Bob each measure their qubit, with the measurement events carefully arranged to be spacelike separated. This means that the events are so far apart and happen so quickly that no signal (even traveling at the speed of light) could communicate the result of one measurement to the other in time.
4. Outcome Comparison: Alice and Bob later meet up and compare their measurement results.

In quantum mechanics, if the qubits were initially in the state $\frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle$, Alice and Bob will always find that their measurement results match. If Alice measures a 0, Bob will also measure a 0; if Alice measures a 1, Bob will also measure a 1. The probability of matching results is 100%.

The Puzzle of Quantum Correlations

This result poses a deep puzzle. How do the two qubits "know" to give matching results if they are spacelike separated? Since no information can travel faster than light, how could the outcome of Alice’s measurement instantaneously affect Bob’s result, or vice versa?

Einstein famously referred to this phenomenon as "spooky action at a distance." He was deeply uncomfortable with the idea that quantum mechanics might involve non-local effects—that is, effects where an action at one location can instantaneously influence an outcome at another, distant location. Moreover, Einstein disliked the inherent randomness in quantum measurements. He believed that the randomness could be a result of some hidden variables that we simply don’t have access to—hidden variables that would determine the outcomes in a deterministic fashion. This idea is encapsulated in his famous quote, "Gott würfelt nicht" ("God does not play dice").

The Idea of Deterministic Local Hidden Variables

To resolve the discomfort with quantum mechanics, one could hypothesize the existence of deterministic local hidden variables:

1. Deterministic: The idea that each qubit “pre-agrees” on its outcome before being separated. For instance, they might agree in advance to both give the result 00 or both give the result 11.
2. Local: The qubits carry this pre-agreed information with them, so no need for faster-than-light communication. The outcome of the measurement is determined locally by the hidden variable.
3. Hidden: The observer cannot access the hidden variables directly. The outcome appears random to the observer because they lack knowledge of these pre-agreed values.

This hypothesis would allow us to preserve a classical, intuitive understanding of the world—one in which there is no spooky action at a distance, and no fundamental randomness.

Einstein, Bohm, and the Search for Hidden Variables

One of the most famous proponents of hidden variable theories was Albert Einstein himself. Throughout his later years, Einstein dedicated significant energy to the pursuit of an alternative to quantum mechanics that would preserve determinism and locality—two principles he held dear. He believed that quantum mechanics, as it was formulated, was incomplete and that the strange phenomena it predicted were the result of our ignorance of underlying hidden variables. Along with his collaborators, Podolsky and Rosen, Einstein formulated the EPR paradox to argue that quantum mechanics must be missing something fundamental. Despite the enormous intellectual effort he invested in this quest, the development of quantum mechanics and subsequent experiments would challenge these views, leading to the formulation of Bell's theorem. Einstein's

reluctance to accept the non-deterministic nature of quantum mechanics highlights the deep philosophical implications of this field and the challenges it posed even to the greatest minds of the 20th century.

Another prominent physicist who explored hidden variable theories was David Bohm. Bohm was deeply influenced by Einstein's dissatisfaction with the indeterminism of quantum mechanics. In 1952, Bohm developed a version of quantum mechanics now known as Bohmian mechanics or the pilot-wave theory, which is a non-local hidden variable theory. Unlike Einstein, who sought a local hidden variable theory, Bohm's theory embraced non-locality but maintained determinism. Bohm's work provided a coherent alternative to the Copenhagen interpretation of quantum mechanics, suggesting that particles have definite positions at all times, guided by a "pilot wave" that evolves according to the Schrödinger equation.

Bohm spent a considerable portion of his career developing and advocating for this theory, despite it being marginalized by the mainstream physics community, which largely favored the probabilistic interpretation of quantum mechanics. His dedication to exploring the foundations of quantum theory and his willingness to challenge the prevailing views contributed to the broader debate about the interpretation of quantum mechanics. Though Bohm's theory is not widely accepted today, it remains an important part of the discussion on the nature of quantum reality and continues to inspire those interested in alternatives to the standard interpretation.

Together, Einstein and Bohm represent two of the most significant figures who championed hidden variable theories, each contributing a unique perspective to the ongoing dialogue about the nature of reality in the quantum world.

Conclusion

In the next article, I will use Python code to illustrate the mind-bending result of Bell, which proves that deterministic local hidden variable theories cannot explain Quantum Mechanics.